The next step in [1]} is to maximally compress the surface \(T_{A_b}\)  just defined towards the side that contains \(B\) . The same difficulty arises: the actual compression disks into the \(B\)  side that one uses involves a choice. On the positive side, the resulting surface \(T_{BA_b}\) , which is well-defined, can be shown to bound a handlebody \(M_{BA_b}\)  in which \(T_{A_b}\)  is a Heegaard surface, providing perhaps an inductive foothold. Furthermore, Powell-like moves (bubble-moves, eyeglass twists, etc) in the Heegaard surface of \(T_{A_b} \subset M_{BA_b}\)  can be 'shadowed' by similar moves in the original splitting surface \(T\) . But again choice is involved in how to shadow the moves. Also, even once a primitive disk is chosen for the splitting \(T_{A_b} \subset M_{BA_b}\)  (inductively we might assume that this choice is unique up to Powell-like moves), such a disk does not pick out a unique primitive disk in our original surface \(T\) .
